Number System Converter

How to use this calculator

Four input fields. Type in any one of them. All others update instantly.

Decimal (base 10) — The number system you use every day. Digits 0–9. Enter any positive integer here and the other three fields populate immediately. This is the most common starting point.

Binary (base 2) — Only digits 0 and 1. The native language of every digital computer. Enter a binary string and the converter shows you the decimal, octal, and hex equivalents. Spaces are ignored — you can type 1111 1111 or 11111111 and get the same result.

Octal (base 8) — Digits 0–7. Less common than the other three but still used in Unix/Linux file permissions and some legacy systems. Enter an octal number to see the equivalent in all other bases.

Hexadecimal (base 16) — Digits 0–9 and letters A–F (or a–f). Hex is everywhere in computing: memory addresses, colour codes, byte representations, debug output. Enter a hex value with or without the 0x prefix.

The result panel below the inputs shows the decimal value prominently, with binary, octal, and hex all displayed together. The bit representation section groups the binary value into nibbles (4-bit groups), shows the total bit count, and tells you how many bytes the value occupies.

The converter supports unsigned integers up to 2⁵³ (JavaScript’s safe integer limit: 9,007,199,254,740,992).

Why four number systems exist in computing

Humans count in base 10 because we have 10 fingers. Computers operate in base 2 because transistors have two states: on and off, 1 and 0. That’s the foundation of everything digital.

Binary works, but reading it is tedious. A 32-bit memory address in binary is 32 ones and zeros with no obvious structure. Hexadecimal solves this by grouping 4 binary digits into a single hex digit. 4 bits = 1 hex digit, so an 8-bit byte becomes 2 hex digits. A 32-bit address becomes 8 hex characters. Suddenly 0xFFAB1234 is readable where 11111111101010110001001000110100 is not.

Octal was popular in older computing systems (6-bit and 12-bit architectures) and survived mainly in Unix file permissions, where three octal digits represent read/write/execute permissions for owner, group, and other. chmod 755 is three octal digits: 7 (rwx), 5 (r-x), 5 (r-x).

The four bases together cover every representation you’ll encounter in modern computing.

Each number system is just a different way of grouping powers of a base. Decimal groups by powers of 10. Binary by powers of 2. Octal by powers of 8. Hex by powers of 16. The value doesn't change, only the notation.

The conversion formulas

Decimal to binary

Repeatedly divide by 2, record the remainder at each step, then read the remainders from bottom to top.

Binary to decimal

Multiply each binary digit by its positional power of 2, sum the results.

Decimal to hexadecimal

Same process as decimal to binary, but divide by 16. Remainders above 9 become A (10), B (11), C (12), D (13), E (14), F (15).

Binary to hexadecimal (the fast way)

Group binary digits into 4-bit nibbles from the right. Convert each nibble to its hex digit directly. No division needed.

Decimal to octal

Divide by 8 repeatedly. Remainders are always 0–7 (octal digits only). Read bottom to top.

Hex to binary (also fast)

Each hex digit expands directly to exactly 4 binary bits. No math needed beyond memorising the 16 nibble values.

Quick reference: decimal 0–16 in all four bases

Decimal	Binary	Octal	Hex
0	0000	0	0
1	0001	1	1
2	0010	2	2
3	0011	3	3
4	0100	4	4
5	0101	5	5
6	0110	6	6
7	0111	7	7
8	1000	10	8
9	1001	11	9
10	1010	12	A
11	1011	13	B
12	1100	14	C
13	1101	15	D
14	1110	16	E
15	1111	17	F
16	10000	20	10

Notice how 16 in hex is written as “10” — just like how decimal 10 means one group of ten. In hex, “10” means one group of sixteen. The carry happens at a different threshold, but the logic is identical.

Bit representation: what the nibble display shows

The bit display groups binary digits into 4-bit nibbles visually. A nibble is the smallest named grouping above a single bit. Two nibbles = 1 byte (8 bits). The display tells you:

Total bit count — The minimum number of bits needed to represent the value. 255 takes 8 bits. 256 takes 9 bits (the 9th bit trips over when you exceed 11111111).

Byte count — Total bits divided by 8, rounded up. Relevant for understanding memory footprint, data type sizing, and network packet fields.

Common bit groupings and their ranges:

Name	Bits	Unsigned range	Hex digits	Used in
Nibble	4	0–15	1	BCD, compression
Byte	8	0–255	2	ASCII, RGB channels, IP octets
Word	16	0–65,535	4	Unicode (BMP), port numbers
Double word	32	0–4,294,967,295	8	IPv4 addresses, uint32
Quad word	64	0–18,446,744,073,709,551,615	16	uint64, timestamps

The reason colour values run 0–255 per channel is that one colour channel fits exactly in one byte. Three channels (RGB) fit in 3 bytes = 6 hex digits, which is why a web colour like #FF5733 is 6 characters: FF (red), 57 (green), 33 (blue).

Where each number system actually appears

Binary in practice

Every boolean value in code is a single bit: true = 1, false = 0. Bitwise operations (AND, OR, XOR, NOT, bit shifting) work directly on binary representations. Subnet masks in networking are often explained in binary: 255.255.255.0 = 11111111.11111111.11111111.00000000.

Hexadecimal in practice

Memory addresses: 0x00007FFF5FBFF8A8 — a 64-bit stack address from a macOS process.

RGB colours: #1E90FF = R:30, G:144, B:255 = dodger blue.

SHA-256 hash: a591a6d40bf420404a011733cfb7b190d62c65bf0bcda32b57b277d9ad9f146e — 64 hex characters = 256 bits = a SHA-256 hash of the word “Hello”.

MAC addresses: 00:1A:2B:3C:4D:5E — 6 bytes written as 6 hex pairs, the hardware address of every network interface.

Octal in Unix permissions

chmod 755 translates directly:

Power of 2 boundaries

The values 128, 256, 512, 1024, 2048, 4096… are all powers of 2. They appear constantly in computing because they represent the exact capacity of n-bit systems.

255 = 2⁸ - 1 (max unsigned byte) 65,535 = 2¹⁶ - 1 (max unsigned 16-bit integer) 4,294,967,295 = 2³² - 1 (max unsigned 32-bit integer)

Real-world examples

Web colour to RGB

A designer sees #3B82F6 in a stylesheet and wants the RGB values.

Reading a memory dump

A developer is debugging a program and sees this hex dump:

48 65 6C 6C 6F 20 57 6F 72 6C 64

IPv4 subnet calculation

Network admin sees subnet 10.0.0.0/8. What’s the broadcast address?

Checking a file’s magic bytes

Every file format has a “magic number” — a specific sequence of bytes at the start of the file that identifies its type. In hex:

Common mistakes

Confusing octal 8 and 9 with decimal. Octal only uses digits 0–7. The number 8 doesn’t exist in octal. If you type 8 or 9 into an octal field, the converter should flag it as invalid. Octal 10 = decimal 8. Octal 11 = decimal 9.

Mixing up hex letter case. A and a both mean 10. FF and ff both mean 255. The converter accepts both. But when comparing values manually — like checking a SHA hash — case differences can cause you to think two identical values differ. Standardise on one case (uppercase for addresses, lowercase for hashes is common).

Forgetting the 0x prefix context. FF as a hex value is 255. Without context, FF could look like two letters. The 0x prefix (0xFF) signals hexadecimal in most programming languages. When writing hex values by hand in code, always include 0x to avoid ambiguity.

Off-by-one in bit counting. The largest value representable in 8 bits is 11111111 = 255, not 256. 256 requires 9 bits (100000000). n bits can represent values from 0 to 2ⁿ - 1. The converter’s bit count display shows the minimum bits needed, which makes this explicit.

Reading binary left to right when converting manually. The least significant bit (rightmost) is position 0 with value 2⁰ = 1. The most significant bit (leftmost) is the highest power. When doing the divide-by-2 method, you read remainders from bottom to top specifically because the first remainder is the least significant bit.

The bottom line

Decimal, binary, octal, and hexadecimal are four ways of writing the same numbers. The converter handles all four simultaneously so you never have to choose which direction to calculate. Type in any field, get all three others instantly.

Binary is the computer’s native representation. Hex is the human-readable shorthand for binary. Octal survives in Unix permissions. Decimal is how the rest of the world communicates numbers.

The bit display at the bottom is the part that actually connects abstract notation to physical reality: it shows you how many transistors (bits) it takes to store the value you typed. 255 takes 8. 256 takes 9. Everything in digital computing comes down to that binary representation at some level, regardless of what notation sits on top of it.